Numerical Method Using Cubic Trigonometric B-Spline Technique for Nonclassical Diffusion Problems

and Applied Analysis 3 Using the approximation of (4), (1) becomes


Introduction
This study deals with the numerical solution of a nonclassical diffusion problem with two nonlocal boundary constraints using cubic trigonometric B-splines.This problem arises in several branches of science.In particular, electrochemistry [1], heat conduction process [2], thermoelasticity [3], plasma physics [4], semiconductor modeling [5], biotechnology [6], control theory, and inverse problems [7].The analysis, development, and implementation of numerical methods for the solution of such diffusion problems have received wide attention in the literature.
Consider an insulated rod of length  located on the axis of the interval [0, ].Let the rod have a source of heat.Let (, ) denote the temperature in the insulated rod with ends held at constant temperatures  1 and  2 , and the initial temperature distribution along the rod is  1 ().The problem is to study the flow of heat in the rod and in this paper the partial differential equation governing the flow of heat in the rod is given by the diffusion equation with specification of energy: where   ,  = 1, 2, 3, 4 are known constants, ,   ( = 1, 2, 3) are known continuous functions,  is the thermal diffusivity of the rod, and ℎ  ( = 1, 2) are prescribed functions.This problem has been studied by Dehghan [8], Martín-Vaquero and Vigo-Aguiar [9], Li and Wu [10], and Golbabai and Javidi [11].Several physical circumstances might be modeled by equation and constraints (1)- (3) and several examples of application in physics with comprehensive derivations of the above mentioned problem can be found in [3,[12][13][14].
There are several numerical methods in the literature that have been developed for solving the proposed problem (1) subject to initial and nonlocal boundary constraints ( 2)- (3).The methods were based, for instance, on the forward Euler method, the backward Euler approach or the Crank-Nicolson scheme [15,16], Laplace transformation [17], and so forth.Dehghan [8] presented four finite difference approaches, namely, the BTCS (backward time centred space) scheme, the implicit (3, 3) Crandall's formula, the 3-point FTCS (forward time centred space) two-level scheme, and the Dufort-Frankel three-level approach for the numerical solution of parabolic equation with nonlocal specification.Martín-Vaquero and Vigo-Aguiar [9] improved the order of convergence of the implicit (3, 3) Crandall's formula proposed by Dehghan [8] and also improved the accuracy of the method.Martín-Vaquero and Vigo-Aguiar [18] developed an algorithm for the solution of the heat conduction equations with nonlocal constraints which reduced the CPU time and enhanced the accuracy of (3, 3) Crandall's formula proposed in [8].Li and Wu [10] proposed an algorithm which was based on the transverse method of lines (TMOL) which can reduce a nonclassical diffusion equation to a series of ordinary differential equations (ODEs).Subsequently, the authors in [10] used an analytic reproducing kernel technique to solve ODEs with integral boundary constraints.Dehghan [19,20], jointly with Tatari and Dehghan [21], has proposed several efficient techniques for the numerical solutions of partial differential equations subject to nonlocal boundary constraints.Golbabai and Javidi [11] introduced a Chebyshev spectral collocation method (CSCM) based on Chebyshev polynomials for solving a parabolic problem subject to nonlocal boundary constraints.For more details on other numerical methods for the solution of a one-dimensional heat equation subject to nonlocal boundary constraints in the literature, see [22][23][24][25][26][27][28][29][30][31].
The study of B-spline functions is a key element in computer-aided geometric design [32][33][34][35].It has also attracted attention in the literature [36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51] to the numerical solution of various differential equations [38][39][40].This is because they have important geometric properties and features that make them amenable to more detailed analysis.Numerical methods based on B-spline functions of various degrees have been utilized for solving initial and boundary value problems.As examples, a cubic B-spline collocation method was used to solve a nonlinear diffusion equation subject to certain initial and Dirichlet boundary constraints [41], a finite element method based on bivariate splines has been used for solving parabolic partial differential equation [42], and the combination of finite difference approach and cubic B-spline method was applied for the solution of a one-dimensional heat equation subject to local boundary constraints [43,44].Goh et al. [45] presented a comparison of cubic B-spline and extended cubic uniform B-spline based collocation methods for solving a one-dimensional heat equation with a nonlocal initial constraint and concluded that extended cubic uniform B-spline with an appropriate value of parameters gives better results than the cubic B-spline.A finite difference scheme based on cubic B-spline was also used for solving the one-dimensional wave equation [43], advection-diffusion equation [44], one-dimensional coupled viscous Burgers' equation [47], system of strongly coupled reaction-diffusion equations [48], and one-dimensional hyperbolic problems [49].
In our present paper, a new two-time level implicit technique is developed to approximate the solution of the nonclassical diffusion problem (1) subject to initial constraints in (2) and nonlocal boundary constraints in (1)-(3).The technique is based on the cubic trigonometric Bspline functions.A finite difference approach and -weighted scheme are applied for the time and space discretization, respectively.Some researchers have considered the ordinary B-spline collocation method for solving the heat equation subject to local and nonlocal boundary constraints but, so far as we are aware, not with the cubic trigonometric Bspline collocation method.Cubic trigonometric B-spline is used as an interpolating function in the space dimension.The unconditional stability property of the method is proved by von Neumann method.The feasibility of the method is shown by test problems with  = ℎ 2 ,  = 1, 2, 4, 5 instead of smaller time step size  = 0.4ℎ 2 and the approximated solutions are found to be in good agreement with the known exact solutions.
The outline of this study is as follows.A numerical solution of nonclassical diffusion problem is presented in Section 2. In Section 3, the cubic trigonometric B-spline is utilized as an interpolating function in the space dimension.The von Neumann approach is used to prove the stability of the method in Section 4. Numerical examples are considered in Section 5 to show the achievability of the proposed method.Finally, the concluding remarks of this study are given in Section 6.
Here the quantities ℎ and  are mesh space size and time step size, respectively.The time derivative can be approximated by using the standard finite difference formula: Using the approximation of ( 4), (1) becomes Using -weighted technique, the space derivatives of (5) can be written as where 0 ≤  ≤ 1 and the subscripts  and  + 1 are successive time levels.It is noted that the system becomes an explicit scheme when  = 0, a fully implicit scheme when  = 1, and a Crank-Nicolson scheme when  = 1/2 [43,49].In this paper, we use the Crank-Nicolson approach.Hence, (6) becomes After simplification, (7) leads to The space derivatives are approximated by using cubic trigonometric B-spline and are discussed in the next section.

Cubic Trigonometric B-Spline Technique
In this section, we discuss the cubic trigonometric B-spline collocation method (CuTBSM) for the numerical solution of the nonclassical diffusion equation (1).Consider a mesh  ≤  ≤  which is equally divided by knots   into  subintervals [  ,  +1 ],  = 0, 1, 2, . . .,  − 1 where  =  0 <  1 < ⋅ ⋅ ⋅ <   = .Our approach for the nonclassical diffusion equation using collocation method with cubic trigonometric B-spline is to seek an approximate solution as [37]  (, ) = −1 where   () are to be determined for the approximated solutions (, ) to the exact solutions (, ), at the point (  ,   ).  () are twice continuously differentiable piecewise cubic trigonometric B-spline basis functions over the mesh defined by [49][50][51]   () Table 1: Values TB  () and its derivatives. where and where ℎ = ( − )/.The approximations    at the point (  ,   ) over subinterval [  ,  +1 ] can be defined as In order to obtain the approximations to the solutions, the values of   () and its derivatives at nodal points are required and these derivatives are tabulated in Table 1, where Using approximate functions (10) and ( 12) and following Mittal and Arora [46], the values at the knots of    and their derivatives up to second order are determined in terms of time parameters    as Substituting ( 12) into (8) gives the following equation: The system thus obtained on simplifying (15) consists of  + 1 linear equations in  + 3 unknowns ) at the time level  =  +1 .Equation ( 9) is applied to the boundary constraints ( 2) and (3) for two additional linear equations to obtain a unique solution of the resulting system: From ( 15), (16), and (17), the system can be written in the matrix vector form as follows: where and  and  are  + 3-dimensional matrix given by where Thus, the system (18) becomes a matrix system of dimension ( + 3) × ( + 3) which is a tridiagonal system that can be solved by the Thomas Algorithm [16].
3.1.Initial State Vector  0 .After the initial vectors  0 have been computed from the initial constraints, the approximate solutions  +1  at a particular time level can be calculated repeatedly by solving the recurrence relation ( 15) [40].
The initial vectors  0 can be obtained from the initial condition and boundary values of the derivatives of the initial condition as follows [40,49]: Thus (22) yields a ( + 3) × ( + 3) matrix system, of the form where The solution of this system can be found by the use of the Thomas Algorithm.

Stability
In this section, the von Neumann stability method is applied for investigating the stability of the proposed scheme.This approach has been used by many researchers [18,40,44,45,[47][48][49]. Substituting the approximate solution  and its derivatives at knots with (, ) = 0, into (6) yields a difference equation with variables   given by Substituting the values of   ,  = 1, 2, 5, 6 into (25) we obtain where )) )) , )) , )) , )) . ( Simplifying it leads to where Now on inserting the trial solutions (one Fourier mode out of the full solution) at a given point   is    =   exp(ℎ) into (28) and rearranging the equations,  is the mode number, ℎ is the element size, and  = √ −1, we get Dividing (30) by    (−2)ℎ and rearranging the equation, we get Let Therefore, (31) can be written as Equation ( 33) can be rewritten as where For stability, the maximum modulus of the eigenvalues of the matrix has to be less than or equal to one [47].Since  > 0,  > 0, and 0 ≤  ≤ 1, we always have Thus, from (36), the proposed scheme for nonclassical diffusion equation (with term, (, ) = 0) is unconditionally stable and it is also unconditionally stable with a general term (, ), since the modulus of the eigenvalues must be less than one [47].We recall Duhamel's principle ( [15], chapter 9); a scheme is stable for equation  ,ℎ V =  if it is stable for the equation  ,ℎ V = 0.This means that there are no constraints on grid size ℎ and step size in time level , but we should prefer those values of ℎ and  for which we obtain the best accuracy of the scheme.

Results and Discussions
In this section, the cubic trigonometric B-spline collocation method is employed to obtain the numerical solutions for one-dimensional nonclassical diffusion problem with nonlocal boundary constraints given in ( 1)-( 3).Two numerical examples are discussed in this section to exhibit the capability and efficiency of the proposed trigonometric spline method.Numerical results are compared with existing methods in the literature and with the exact solution at the different nodal points   for some time levels   using some particular space step size ℎ and time step .In order to calculate the maximum errors and relative  2 error norms of the proposed method numerically, we use the following formulas: This test problem is from Dehghan [8] and Li and Wu [10] and the known solution is (, ) =  −(+sin ) .We compare the maximum errors with TMOL [10] when they are considered with space size ℎ = 0.05 and errors are recorded with several values of time step , given in Table 2 and also shown in Figure 1.It is worth noting that the results obtained using CuTBSM are more accurate as compared to TMOL [10].We also compare the relative errors of numerical value of (0.6, 0.1) with different space step ℎ = 0.05, 0.025, 0.01 and with time step size  = ℎ 2 ,  = 1, 2, 4, 5, instead of  = 0.4ℎ 2 which was used in the BTCS [8], the implicit (3, 3) Crandall's formula [8], the 3-point FTCS [8], and the Dufort-Frankel three-level approach [8] and TMOL [10], and these results are tabulated in Table 3.It is clearly shown from this table that the obtained results by using CuTBSM are more precise as compared to methods in [8,10].

Concluding Remarks
In this paper, a new two-time level implicit scheme based on cubic trigonometric B-spline has been used to solve the nonclassical diffusion problem with known initial and with nonlocal boundary constraints instead of the usual boundary  constraints.A usual finite difference discretization is used for time derivatives and cubic trigonometric B-spline is applied for space derivatives.It is noted that the accuracy of solution may reduce as time increases due to the time truncation errors of time derivative term [47].The cubic trigonometric B-spline method used in this paper is simple and straightforward to apply.An advantage of using the cubic trigonometric B-spline method outlined in this paper is that it produces a spline function on each new time line which can be used to obtain the solutions at any intermediate point in the space direction whereas the finite difference approach yields the solution only at the selected points.The CuTBSM has approximated the solution with more accurate results for time step size  = ℎ 2 ,  = 1, 2, 4, 5, as compared to some finite difference schemes with smaller time step size  = 0.4ℎ 2 such as BTCS, Crandall's formula, FTCS, the Dufort-Frankel scheme, and TMOL based on reproducing kernel.The proposed method is shown to be unconditionally stable.
It is also evident from the examples that the approximate solution is very close to the exact solution.

3 Figure 1 :x 5 Figure 2 :
Figure 1: Absolute errors of numerical values at different time levels for Example 1 by using ℎ = 0.05 and  = 0.001.

Figure 4 :
Figure 4: A comparison of numerical solution and known solution at different time levels for Example 2 with ℎ = 0.05 and  = 0.01.

Figure 5 :
Figure 5: Absolute errors of numerical values at  = 1 for Example 3 with ℎ = 0.01 and different time step size.

Figure 6 :
Figure 6: A comparison of numerical solution and known solution at different time levels for Example 3 with ℎ = 0.05 and  = 0.01.

Table 2 :
[10]lute errors for Example 1 with ℎ = 0.05 and several values of time step by present method (CuTBSM) and compare with TMOL[10].

Table 3 :
Relative errors of numerical method at  (0.6, 0.1) for Example 1 with  = 0.4 ℎ 2 and various values of space step.

Table 4 :
[10]lute errors for Example 2 with ℎ = 0.05 and several values of time step by present method and TMOL[10].

Table 5 :
[11]lute errors of numerical value of  for Example 2 with different space step size,  = 0.0001, at selected time levels and compare with CSCM[11].

Table 6 :
Relative errors of numerical method at  (0.6, 1.0) for Example 2 with various values of space step.

Table 7 :
Relative errors of numerical method at  (0.6, 1.0) for Example 3 with different time step size and various values of space step.Exact and Approx.Sol at t = 0.1 Exact and Approx.Sol at t = 0.3 Exact and Approx.Sol at t = 1.0 Exact and Approx.Sol at t = 0.9 Exact and Approx.Sol at t = 0.7 Exact and Approx.Sol at t = 0.5